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Mirrors > Home > ILE Home > Th. List > xrrebnd | GIF version |
Description: An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
xrrebnd | ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9563 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
3 | mnflt 9569 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
4 | ltpnf 9567 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | 3, 4 | jca 304 | . . . 4 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
6 | 2, 5 | 2thd 174 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
7 | renepnf 7813 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
8 | 7 | necon2bi 2363 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ) |
9 | pnfxr 7818 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9566 | . . . . . . 7 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ ¬ +∞ < +∞ |
12 | breq1 3932 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 664 | . . . . 5 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 13 | intnand 916 | . . . 4 ⊢ (𝐴 = +∞ → ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
15 | 8, 14 | 2falsed 691 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
16 | renemnf 7814 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
17 | 16 | necon2bi 2363 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
18 | mnfxr 7822 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
19 | xrltnr 9566 | . . . . . . 7 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ ¬ -∞ < -∞ |
21 | breq2 3933 | . . . . . 6 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
22 | 20, 21 | mtbiri 664 | . . . . 5 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
23 | 22 | intnanrd 917 | . . . 4 ⊢ (𝐴 = -∞ → ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
24 | 17, 23 | 2falsed 691 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
25 | 6, 15, 24 | 3jaoi 1281 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
26 | 1, 25 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 +∞cpnf 7797 -∞cmnf 7798 ℝ*cxr 7799 < clt 7800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 |
This theorem is referenced by: xrre 9603 xrre2 9604 xrre3 9605 elioc2 9719 elico2 9720 elicc2 9721 xblpnfps 12567 xblpnf 12568 |
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